2008/11/25 by Lizhen Ji, Ji, Lizhen, Enrico Leuzinger +1
Mathematics · #14H42 #32G15 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT) #math.DG #math.GT #msc:14H42 #msc:32G15
paper · pdf · doi:10.48550/arxiv.0811.4059
arxiv created 2008/11/25 · arxiv updated 2009/12/01
Let \mathcal Mg denote the moduli space of compact Riemann surfaces of genus g and let \mathcal Ag be the space of principally polarized abelian varieties of (complex) dimension g. Let J:\mathcal Mg\longrightarrow \mathcal Ag be the map which associates to a Riemann surface its Jacobian. The map J is injective, and the image J(\mathcal Mg) is contained in a proper subvariety of \mathcal Ag when g≥ 4. The classical and long-studied Schottky problem is to characterize the Jacobian locus \mathcal Jg:=J(\mathcal Mg) in \mathcal Ag. In this paper we adress a large scale version of this problem posed by Farb and called the \em coarse Schottky problem: How does \mathcal Jg look "from far away", or how "dense" is \mathcal Jg in the sense of coarse geometry? The coarse geometry of the Siegel modular variety \mathcal Ag is encoded in its asymptotic cone \textupCone_∞(\mathcal Ag), which is a Euclidean simplicial cone of (real) dimension g. Our main result asserts that the Jacobian locus \mathcal Jg is "asymptotically large", or "coarsely dense" in \mathcal Ag. More precisely, the subset of \textupCone_∞(\mathcal Ag) determinded by \mathcal Jg actually coincides with this cone. The proof also shows that the Jacobian locus of hyperelliptic curves is coarsely dense in \mathcal Ag as well. We also study the boundary points of the Jacobian locus \mathcal Jg in \mathcal Ag and in the Baily-Borel and the Borel-Serre compactification. We show that for large genus g the set of boundary points of \mathcal Jg in these compactifications is "small".